From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/429 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Signed associahedra Date: Tue, 8 Jul 1997 14:21:00 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016960 25711 80.91.229.2 (29 Apr 2009 14:56:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:56:00 +0000 (UTC) To: categories Original-X-From: cat-dist Tue Jul 8 14:21:01 1997 Original-Received: by mailserv.mta.ca; id AA29688; Tue, 8 Jul 1997 14:21:00 -0300 Original-Lines: 64 Xref: news.gmane.org gmane.science.mathematics.categories:429 Archived-At: Date: Tue, 8 Jul 1997 12:47:46 -0400 (EDT) From: James Stasheff Reiner and Burgiel's recent paper provoked the questions at the beginning and end of the following: ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 ---------- Forwarded message ---------- Date: Tue, 8 Jul 1997 11:33:31 -0500 (CDT) From: Victor Reiner To: jds@math.upenn.edu Cc: burgiel@math.uic.edu Subject: Signed associahedra Dear Jim, > Only thing more I could ask is to have the tree descrition > in addition to the triangulation description of the cells. I think we can oblige. A signed dissection of the (n+2)-gon corresponds to a plane tree T having n+1 leaves in which one assigns +,-, or 0 to every "nook" between two branches of the tree (think of the +,-,0's as being like lint trapped between ones toes!). For example, a vertex having 5 children in the tree will have 4 nooks between its branches, and hence require 4 choices of +,-,0. Furthermore, whenever any of the nooks below some vertex are assigned 0, all of the nooks below that vertex must be assigned 0. Then a signed dissection of the first kind (which indexes the cells in the simple signed associahedron) requires that only the root vertex can have its nooks assigned 0. And a signed dissection of the second kind (which indexes the cells in the non-simple signed associahedron) requires that only the vertices whose children are all leaves can have their nooks assigned 0. The partial order, roughly speaking, translates into contracting non-leaf edges in the tree and comparing the +,-,0 assignments in the nooks. Do you know of any category/homotopy theoretic applications for either of these signed associahedra? Best wishes, Vic Reiner